The relationship between monopole harmonics and spin-weighted spherical
harmonics
Tevian Dray
Instituut voor Theoretische Fysica, Princetonplein 5, Postbus 80. 006, 3508 TA Utrecht, The Netherland1'} and
School ofMathematics, Institute for Advanced Study, Princeton, New Jersey 08540
(Received 1 October 1984; accepted for publication 28 December 1984)
We compare two independent generalizations of the usual spherical harmonics, namely
monopole harmonics and spin-weighted spherical harmonics, and make precise the sense in
which they can be considered to be the same. By analogy with the spin-gauge language, raising
and lowering operators for the monopole index of the monopole harmonics can immediately be
written down.
I. INTRODUCTION
Once again physicists in two completely different areas
have independently developed the same mathematics. Wu
and Yang! introduced 2 monopole harmonics as particular
solutions of the Schrodinger equation for an electron in the
field of a Dirac magnetic monopole. Newman and Penrose3
introduced 2 spin-weighted spherical harmonics as a means
to describe certain quantities exhibiting a particular "spingauge" behavior which occur naturally in the asymptotic
expansion of the gravitational field in null directions.
In what follows we compare these two generalizations
of the usual spherical harmonics and show that, for a particular choice of spin gauge, the spin-weighted spherical harmonics reduce to the monopole harmonics. As a simple application of this result, we note that the fundamental
operators in the spin-gauge language raise or lower the spin
weight by 1. Thus, writing these operators in the appropriate
gauge immediately yields operators which raise or lower the
monopole index of the monopole harmonics by 1. Going in
the other direction, we adapt the angular momentum operators of the Schrodinger picture to the spin-gauge language
and derive the corresponding operators there.
In Sec. II we first review monopole harmonics and in
Sec. III we do the same for spin-weighted spherical harmonics. We compare the two in Sec. IV and then discuss our
results in Sec. V.
low follows Wu and Yang l we will deliberately deemphasize
the underlying fiber bundle structure.
Define the regions Ra and Rb on the sphere by
= {O";O<1Tj,
=
{O<O";1Tj.
(1)
The relevant Schrodinger equation is
[-(lIr)a,(ra,)+(lIr)[L 2 - q2] + V-E]f/!=O, (2)
Ra
Rb
where V(r) is the potential, E is the energy eigenvalue, L 2 is
the total angular momentum operator, and q = eg (see Ref.
8).
One makes the ansatz
f/!(r,O,rp) = R (r)Yqlm(O,rp),
(3)
where the Yq1m are characterized by their angular momentum eigenvalues
L 2Yq1m = 1(1
+ I)Yq1m ,
LzYq1m = mYq1m .
(4a)
We also have
L± Yq1m = [(l =Fm)(1 + 1 =Fm)] l12Yqlm± l '
(4b)
The fiber bundle structure can be interpreted as follows: The
angular momentum operators take different forms in regions
R a and R b, leading to different functions Y: 1m and Y!/m
which together make up a monopole harmonic Yq1m . In this
paper, however, we will only be concerned with the/unctions
Y:/m and Y!/m'
The angular momentum operators are
II. MONOPOLE HARMONICS
The term "monopole harmonics" was first used by Wu
and Yang l to describe solutions of the SchrOdinger equation
for an electron in the field of a magnetic monopole. However, the functions used in this description are almost as old
as the relevant Schrodinger equation itself, which dates back
to the original paper on monopoles by Dirac. 4
These functions were first discussed by TammS and
Fierz6 and then by numerous other authors. 7
The fundamental difference in the approach ofWu and
Yang l is that elements of their Hilbert space are not functions at all, but rather sections of a particular fiber bundle.
This eliminates the string singularity of the original description of the Dirac monopole. Although the presentation bea)
Permanent address.
1030
J. Math. Phys. 26 (5), May 1985
= -41 +
L! =
L b±
- i
a",
.~
sin 0
+ q,
(cosO-I)L:,
(5c)
(5d)
= e±ltp( ± ae + i ~s 0 a
smO
'"
_ q(I ~ cos 0)),
(5e)
smO
2'
2q2
(L 2)b= -41 -~(1 + cos O)a", +-'-2-(1 +cosO)
smO
smO
= -41
0022-2488/85/051030-04$02.50
+
.~
sm 0
(cosO + I)L!,
@) 1985 American Institute of Physics
(5f)
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where
(14)
L1 =a 2 + cosO a +_I_a 2
()
sin 0 ()
sin 2 0 '"
a
is the Laplace operator on the two-sphere.
With appropriate normalization the Y qlm satisfy
i
YqlmYq/,m' dS=811'8 mm"
(6a)
where the integral is over the full two-sphere; we note that
the integrand is the same in the regions R a and R b. We also
have
(6b)
Y qlm
= 0,
for I<
Iql,
(6c)
where the Y lm denote the usual spherical harmonics. Finally, we note that {Yqlm J for given q is complete in the following sense: Given any sectionf = (fa I b), wherefa andfb are
functions on R a and R b, respectively, satisfyingfa = e2qi"'f b ,
thenf can be expanded as a linear combination ofthe Y qlm .
We can, however, combine m and 7i into operators which
raise or lower the spin weight. For sW(Q ) = s define lO
3Q=m a aaQ + 27isQ, d=maaaQ-2asQ,
(15)
where 3 is the Icelandic letter "edth"; note that 3Q is the
complex conjugate of 3Q since sW(Q ) = - sw(Q). The fundamental property of these operators is
sw(3Q) =s + 1,
sw(Q) =s=> sw('3i'Q)
_ s - 1,
u
i.e., sw (3) = 1, sw (d) = - 1. We also have
[3,d] Q =
Newman and Penrose3 introduced spin-weighted
spherical harmonics based on ideas in Janis and Newman 9 in
order to describe the asymptotic behavior of the gravitational field of isolated systems at large null distances from
the source. Although they did this for a particular choice of
spin gauge (the "standard" spin gauge) the concept can be
immediately generalized to an arbitrary spin gauge. Except
for this minor difference our presentation follows Newman
and Penrose. 3
Consider a two-sphere with the usual metric
gab dx a dx b = dO 2 + sin 2 0 dql.
(7)
Instead of the usual orthonormal basis [a(),(lIsin 0 )a",], we
introduce a complex null basis (ma,m a) via
(8)
where the bar denotes complex conjugation. The general m
can thus be written
a
(9)
The choice of the function r( O,tp) will be called the choice of a
spin gauge. We are thus led to consider transformations of
the form
(10)
A quantity Q whose behavior under this gauge transformation is
(17)
2sQ.
The standard gauge is given by choosing r = 0 in (9), thus
= a() + (ilsin 0 )a", - s(cos Olsin 0),
do = a() - (ilsin 0 )a", + s(cos Olsin 0).
30
(18)
In an arbitrary gauge we have
:II
ir[a()
u=e
III. SPIN-WEIGHTED SPHERICAL HARMONICS
-
(16)
1)]'
cos O .
1)]
sin 0 + zr,() + sin Or.", .
i a +s( --.
cos--zr,()
O . +-.-r,
+-.-
d = e - ir[a()
sm 0 '"
sm 0
i a
(
- sin 0 '" - s -
sm 0 '"
(19)
We can now obtain the spin-weighted spherical harmonics
(for integer spin) s Y lm by raising and lowering the spin
weight of the usual spherical harmonics Ylm(O,tp)
[sw(Ylm ) = 0] II
(I - s)! ] 112 s
-3 Ylm , O':;'s.:;.l,
[(I +s)!
(I + s)! ] 112 (
[ (/-s)!
0,
_
I)Sd - sy
-1.:;.s.:;.O, (20)
1m'
I< lsi.
We summarize the properties of the s Ylm
sw(s Y lm ) = s,
3(sYlm ) =
d(sYlm ) =
(2Ia)
+ [(l-s)(/+s+
-
I)]1I2 s + I Y lm '
(2Ib)
[(l+s)(/-s+ I)]1I2 s _ I Y lm '
(2Ic)
oYlm = Y lm ,
(2Id)
lsYlm sYI'm' dS=8 11 ,8mm ,.
(2Ie)
We can ask if there are generalizations ofthe usual angular
momentum operators, i. e., operators L z ' L ± ,L 2 satisfying
[cf. (4)]
(11 )
is said to have spin weight s [sw(Q ) = s]. The simplest example of this is
sw(ma) = + 1, sw(m a) = - 1.
(12)
Note that not all quantities have a well-defined spin weight.
An example of this is
(13)
where Va denotes covariant differentiation on the twosphere, which transforms under (10) as
1031
J, Math, Phys., Vol. 26, No, 5, May 1985
L± sYlm = [(l =t=m)(1 + 1 ± m)]112 sYlm ±1>
L 2 s Y lm = I (I + 1) s Y lm .
(22)
Since these imply that
[L,3] = 0 = [L,d],
(23)
whereL represents any of the angular momentum operators,
one can easily solve for these operators. The result is 12
L z = - i a",
- sr.""
Tevian Dray
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'3B=e+ i",(ae __._i_a +q (1 +.COS())).
L ± -e
- ±iq>[ ±ae
+.
icos()
() a'"
sm () '"
sm ()
[To obtain the correct normalization merely divide these by
the constant on the right side of(21b) or (21c) withs = q.]
sm
1
.
cos ()
+ s( - sin () + 'r.e + sin () r.",
)]
'
L2= -.1'+ 2scos() L +~
sin2 ()
V. DISCUSSION
sin 2 ()
Z
= -.1 + is(.1r) - 2sr.e(Ly cos", - Lx sin",)
~
+ ~() cos
sm
()
-
r.", )L
r(.r.e
.2
Z
-
+
y'",. -2 ()1 ) ,
sm
where
2
.1 ' =.1 - is(.1r) - 2is(r.e ae + r.", a",/sin ())
-r(Y.e
+ y'",/sin2 (),
and
Our result (26) should not be surprising. The monopole
harmonics are analytic, whereas the operator 3 0 has a direction-dependent limit at () = 0 and () = 11". Going to the gauge
A or B is necessary in order to tum 3 into an analytic operator on the region R 0 or R b!14
Futhermore, since the. Y1m of course have spin weight
s, our result can be interpreted as follows: Remove the explicit q dependence (i. e., e ± Iqtp) from the q Yi::. The result is
precisely the spin-weighted spherical harmonics q Y~m in
standard gauge.
(24)
Here,.1 ' is just the operator obtained from.1 by the substitutions
a",f-+a", -isr.""
aef-+ae -isr.e·
Note that in the standard gauge, den~ted "0" the L o±
are just the angular momentum operators J ± given in Landau and Lifschitz l3 for the symmetric top (with k there identified with - s here). The similarity between the symmetric
top operators and the Yqlm has already been pointed out,
e. g., in Ref. 6.
We have only explicitly treated the spin-weighted
spherical harmonics for integer spin. However, the argument used in Sec. III to introduce the angular momentum
operators L can be inverted: we could equally well define the
spin-weighted spherical harmonics as eigenfunctions of L. It
is then obvious that the results of Sec. IV are also valid for
half-integer spin.
Note added: In fact, if we let. yrm denote the spinweighted spherical harmonics in spin gauge r [Eq. (9)] then II
(28)
_ ( - 1)'(21
=
IV. COMPARISON OF MONOPOLE AND SPINWEIGHTED SPHERICAL HARMONICS
Comparing (24) with (5) we see that if we introduce the
gauges A, defined by r = + rp, and B, defined by r = - rp
[in (9)], and if we make the identification q = s, then
(25)
where L again represents any of the angular momentum operators. But since the Yqlm are fully determined up to a constant phase factor for each q by specifying q, the behavior of
the angular momentum operators [Eq. (4)], and the normalization condition (6a), and since the. Y lm have the same behavior with respect to angular momentum [Eq. (22)] and the
same normalization [Eq. (21e)], we see that Yqlm and qYlm
differ at most by a constant (q-dependent) phase factor. With
our. Ylm as defined in (20) we have
_
yA.
yb _ yB
(26)
Y oqlm -q
1m'
qlm -q 1m'
This is our main result.
Note that we can now immediately give raising and
lowering operators for the monopole index of the monopole
harmonics; these are just 3 and '3 in the appropriate gauge:
3A.=e+iq>(ae +_._i_a +q (1-.COS())),
sm () '"
3B =e-iq>(a +_i_a _
e
'()'"
sm
sm ()
q
(1 +COS()))
.()
,
sm
(27)
'3A.=e- I"'(a _~ _
e
1032
'()'"
sm
q
(I-COS()))
.()
,
sm
J. Math. Phys.• Vol. 26. No.5, May 1985
+ 1)1/2
(411")1/2
I
D -.m(rp,(),r),
where the D I_.m are the Wigner D functions as given by
Goldberg et al. IS Thus, choosing a gauge rin the sense of this
paper corresponds to fixing a Euler angle ( - r) in the argument of the Wigner D functions. As pointed out by the referee, the spin-weighted spherical harmonics in standard
gauge s Y~m and the monopole harmonics Y;i!:. merely correspond to different choices of this Euler angle.
ACKNOWLEDGMENTS
I would like to thank Professor Chen Ning Yang for
providing Ref. 7, Peter Batenburg for bringing the symmetric top (Ref. 13) to my attention and for pointing out some
mistakes in the original manuscript, Malcolm Perry and
Annti Niemi for discussions on the Wigner D functions, and
the referee for suggesting Ref. 15.
This work was supported in part by the Stichting voor
Fundamenteel Onderzoek der Materie.
IT. T. Wu and C. N. Yang, Nucl. Phys. B 107, 365 (1976).
2In both cases the original ideas can be traced to earlier work. This is discussed in Sees. II and III below.
3E. T. Newman and R. Penrose, J. Math. Phys. 7, 863 (1966).
·P. A. M. Dirac, Proc. R. Soc. London Ser. A 133, 60 (1931).
SIg. Tamm, Z. Phys. 71, 141 (1931).
6M. Fierz, Relv. Phys. Acta 17, 27 (1944).
7p. P. Banderet, Relv. Phys. Acta 19, 503 (1946); Rarish-Chandra, Phys.
Rev. 74, 883 (1948); A. S. Goldhaber, Phys. Rev. 140, BI407 (1965).
Tevian Dray
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8We have set Ii = c = 1; e is the electron charge and g the strength of the
monopole.
9A. I. Janis and E. T. Newman,J. Math. Phys. 6,902 (1965).
l'7hese differ by a factor of ( - 1) from the operators defined in Ref. 3.
IIThese s Y 1m differ from those in Ref. 3 by a factor ( - 1)'. The reason for
this choice will become clear in Sec. IV.
12These operators reduce to the standard ones for s = O. Note that strictly
1033
J. Math. Phys., Vol. 26, No.5, May 1965
speaking "s" must be interpreted as an operator.
13L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon, New
York, 1977), 3rd ed., p. 417; see also F. Reich and H. Rademacher, Z.
Phys. 39, 444 (1926).
I~is idea is implicit in Ref. 3.
15J. N. Goldberg, A. J. Macfarlane. E. T. Newman, F. Rohrlich, and E. C.
G. Sudershan, J. Math. Phys. 8, 2155 (1967).
Tevian Dray
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